Digital Communication Systemssloyka/elg4179/Lec_6_ELG...• Band-limited systems and intersymbol...

ELG4179: Wireless Communication Fundamentals © S.Loyka

Lecture 6 10-Oct-18 1(35)

Digital Communication Systems

Review of ELG3175:

• Fourier Transform/Series • Spectra • Modulation/demodulation, baseband/bandpass signals • Digital signaling • Intersymbol interference (ISI)

Use your notes/textbook and the Appendix.

Generic modulation formats:

• AM, PM and FM • properties

Why digital?

• better spectral efficiency • better noise immunity (quality) • allows DSP (not expensive)



Block diagram of a digital

communication system

P.M. Shankar, Introduction to Wireless Systems, Wiley, 2002.

(Tx)

(Rx)



Digital Communications

Digital source: select a message m from a set of possible

messages { }1 2, ......... Nm m m (alphabet).

Digital modulator: each message im is represented via a

corresponding (time-limited) symbol ( )is t , 0 st T≤ ≤ .

( )i i

m s t→ , 0 t T≤ ≤ (7.1)

Transmitted sequence:

( ) ( )ii

x t s t iT= −∑ , (7.2)

Pulse-amplitude modulation (PAM): ( ) ( )i is t a p t= ,

( ) ( )ii

x t a p t iT= −∑ (7.3)

i.e. each message is encoded by amplitude ia :

i im a→ .

Baseband system: each ( )is t or ( )p t is a baseband pulse.

Bandpass (RF) system: pulses are bandpass (or RF).



Baseband Digital Signaling

Also known as “line coding”: each ( )is t or ( )p t is a baseband

pulse.

Examples:

• Binary data representation: 0 or 1. • Unipolar modulation: high level (e.g., 5V) / zero, or 1/0 • Bipolar modulation: +high level / -high level, or +1/-1

1bit (data) rate [bit/s]

b

RT

= =

L.W. Couch II, Digital and Analog Communication Systems, Prentice Hall, 2001.

( ) ( )ii

x t a p t iT= −∑



Spectral Characteristics

Spectrum is a very limited and expensive resource in wireless

communications.

Wireless systems are band-limited.

How to select ( )p t ?

• Rectangular pulse, • Sinc pulse, • Raised cosine pulse, • Others.



Example: Rectangular Pulse

2

T−

2

T

1

T

1

T−

T

( )x

S f

f

t

signal

spectrum

( )1, / 2

( )0, / 2

t Tx t t

t T

( )sin

sinc( )x

fTS f T T fT

fT

π

= =

π

1

( ) ( ) 2j ftxS f x t e dt+∞

− π

−∞

= ∫

?f∆ =




1R

T=

( ) 2 2sinc ( )P f A T fT=

( )2 2

2sinc ( ) ( )2 2

A T AP f fT f= + δ

?f∆ =



Digital Signaling over Bandlimited Channels: Pulse Shaping and ISI

Practical channels are band-limited -> pulses spread in time and are smeared into adjacent

slots. This is intersymbol interference (ISI).



Transmission over a Band-Limited Channel

baseband transmission system

Input PAM signal:

( ) ( )n Tnx t a s t nT= −∑ (7.4)

Output signal:

( ) ( ),n Rny t a s t nT= −∑ (7.5)

where ( ) ( )* ( )* ( )* ( ) ( )* ( )R T T C R Ts t s t h t h t h t s t h t= = is

the Rx symbol.

Sampled (at t=mT ) output ( )my y mT= :

0 ,m n m n m n m nn n m

y a s a s a s− −

≠= = +∑ ∑ (7.6)

where T = symbol interval, ( )m Rs s mT= .


Lecture 6 10-Oct-18 10(35)

Pulse Shaping to Eliminate ISI

• Nyquist (1928) discovered 3 methods to eliminate ISI: – zero ISI pulse shaping,

– controlled ISI (eliminated later on by, say, coding),

– zero average ISI (negative and positive areas under the pulse in an adjacent interval are equal).

• Zero ISI pulse shaping:

( ) 0

0 0

, 0

0, 0

m m n m n m mn m

s ns nT

n

y a s a s y a s−

≠

== ⇒

≠

= + ⇒ =∑

(7.7)

• Pulse shapes: - rectangular pulse

- sinc pulse

- raised cosine pulse

Example: sinc pulse,

( ) sinc( ) ( ) sinc( ) 0, 0s t Rt s nT n n= ⇒ = = ≠

where R=1/T = transmission (symbol) rate [symbols/s].


Lecture 6 10-Oct-18 11(35)

Zero ISI: sinc Pulse

3 2 1 0 1 2 30.3

0

0.3

0.6

0.9

sinc(t) pulse

t

( ) sinc( ) ( ) sinc( ) 0, 0s t Rt s nT n n= ⇒ = = ≠ ,

1/R T= = transmission rate [symbols/s].

Sinc pulse allows to eliminate ISI at sampling instants.

However, it has some (2) serious drawbacks.


Lecture 6 10-Oct-18 12(35)

Nyquist Criterion for Zero ISI

• Provides a generic solution to the zero ISI problem. • A necessary and sufficient condition for zero ISI:

s

m

mS f T

T

∞

=−∞

+ =

∑ (7.8)

maxW F=

Consider a channel badndlimited to [0,Fmax]. There are 3

possible cases:

1) max

1/ 2T R F= > � no way to eliminate ISI.

2) max

2R F= � only sinc( )Rt eliminates ISI. The highest

possible transmission rate f0 for transmission with zero ISI is

2Fmax , not Fmax (what a surprise!)

3) max

2R F< � many signals may eliminate ISI in this case.

Sinc pulse: max

1

2 2

Rf F

T∆ = = = . Q: rectangular pulse?


Lecture 6 10-Oct-18 13(35)

The Raised Cosine Pulse

When max

2R F< , raised cosine pulse is widely used.

Its spectrum is

( )

1, 0

2

1 1 11 cos ,

2 2 2 2

10,

2

rc

T f R

T TS f f R R f R

f R

−α≤ ≤

π −α −α +α

= + − ≤ ≤ α + α

>

(7.9)

where 0 1≤ α ≤ is roll-off factor,

The bandwidth above the Nyquist frequency f0/2 is called excess

bandwidth.

Time-domain waveform (shape) of the pulse is

( ) ( )( )

2 2 2

cos /sinc /

1 4 /

t Ts t t T

t T

πα

=

− α

(7.10)

Baseband bandwidth = ?

RF (bandpass) bandwidth = ?


Lecture 6 10-Oct-18 14(35)

Note that: (1) 0α = , pulse reduces to ( )sinc Rt and 0 max2R f F= = (2) 1α = , symbol rate

maxR F=

(3) tails decay as t3 -> mistiming is not a big problem

(4) smooth shape of the spectrum -> easy to design filter

1 0.5 0 0.5 10

0.5

1

alfa=1

alfa=0.5

alfa=0

Raised Cosine Spectrum

f /f0

3 2 1 0 1 2 3

0

0.5

1

alfa=1

alfa=0.5

alfa=0

Time-Domain pulse

t/T


Lecture 6 10-Oct-18 15(35)

Bandpass (RF) Digital Signaling

How to transform the baseband signal to the RF frequencies (for

efficient wireless transmission)?

Most popular approach: combine the baseband signaling with the

DSB-SC modulation:

( ) ( ) cos( )c c

x t s t A t= ⋅ ω (7.11)

i.e. multiply the digitally-modulated message (baseband)

( ) ( )ii

s t a p t iT= −∑ with the carrier cos( )c cA tω .

Compare to the analog DSB-SC:

( ) ( ) cos( )c c

x t m t A t= ⋅ ω (7.12)

where ( )m t is the message: (7.11) = (7.12) when ( ) ( )m t s t= , i.e.

( )s t serves as an analog message to DSB-SC.

Spectrum of the modulated bandpass signal:

( ) ( ) ( )2

c

x m c m c

AS f S f f S f f = − + + (7.13)

f

( )m

S f

cf0

f

( )x

S f

cf

cf−

mSmS

0


Lecture 6 10-Oct-18 16(35)

Binary Phase Shift Keying (BPSK)

Assume rectangular pulse shape and random binary message:

( ) ( )ii

s t a p t iT= −∑ , 1ia = ± , ( ) ( )p t t= Π (7.14)

Then, the PSD of the modulated signal ( ) ( ) cos( )c c

x t s t A t= ⋅ ω is

( )2

2sinc (( ) )2

c

BPSK c

A TP f f f T= − (7.15)

Bandwidth ?f∆ =

RF vs. baseband badwidth?


Lecture 6 10-Oct-18 17(35)

Digital DSB-SC Transmitter

source BPFBPF

carriercarrier

baseband

modulator

baseband

modulator×

cos(2 )c c

A f tπ

{ }im ( )s t ( )x t

Digital baseband modulation: i im a→ , ( ) ( )ii

s t a p t iT= −∑

Baseband (digitally-modulated) signal: ( )s t

Bandpass (RF) modulated signal: ( ) ( ) cos( )c c

x t s t A t= ⋅ ω

BPF: eliminates out-of-band emission

* Demodulation (detection) ?


Lecture 6 10-Oct-18 18(35)

Summary

• Review of digital communications.

• Baseband digital modulation (line coding).

• Band-limited systems and intersymbol interference.

• Bandpass (RF) modulation.

• Spectra.

• Modulators/demodulators (detectors), block diagrams.

• Review of FT & modulation from ELG3175.

Reading:

� Rappaport, Ch. 6 (6.1-6.10).

� L.W. Couch II, Digital and Analog Communication Systems, 7th Edition, Prentice Hall, 2007. (other editions are OK as well)

� Other books (see the reference list).

Note: Do not forget to do end-of-chapter problems. Remember

the learning efficiency pyramid!


Lecture 6 10-Oct-18 19(35)

Appendix 1:

Baseband/Bandpass Signals and Modulation

In what follows is a review of baseband/bandpass signals and

modulation (DSB-SC), corresponding modulators and detectors

as well as basic digital modulation techniques. This material was

covered in ELG3175 and you are advised to consult your notes

and the textbook of that course.

A good textbook to follow is this:

L.W. Couch II, Digital and Analog Communication Systems, 7th

Edition, Prentice Hall, 2007. (other editions are OK as well)


Lecture 6 10-Oct-18 20(35)

Baseband & Bandpass Signals

• Baseband (lowpass) signal: spectrum is (possibly) nonzero

around the origin (f=0) and zero (negligible) elsewhere:

• Bandpass (narrowband) signal: spectrum is (possibly) nonzero

around the carrier frequency fc

and zero (negligible) elsewhere:

Lecture 5

max( ) 0,

xS f f f= >

( ) 0,x c

S f f f B= − >

f∆

f

( )x

S f

f∆

f

( )x

S f

cf


Lecture 6 10-Oct-18 21(35)

DSB-SC: General Case

� DSB-SC signal:

� Spectrum:

� What do you see on a spectrum analyzer?

� Bandwidth ? Power efficiency? PSD?

( ) ( ) ( )2

c

x m c m c

AS f S f f S f f= − + +

( ) ( ) ( )cos 2c c

x t A m t f t= π

f

( )m

S f

cf0

f

( )x

S f

cf

cf−

mSmS

0

Lecture 6


Lecture 6 10-Oct-18 22(35)

Generation of DSB-SC

� Generation:

� Mixer. Not practical in many cases.

� Filtered conventional AM. Not practical.

� Balanced modulator:

J.Proakis, M.Salehi, Communications Systems Engineering, Prentice Hall, 2002

Lecture 6


Lecture 6 10-Oct-18 23(35)

Demodulation of DSB-SC

� Demodulation - Costas loop:

Lecture 6


Details: Couch,

Sec. 4.14, 5.4

0?

/ 2?

?

θ =

= π

= π

Why 2 channels?


Lecture 6 10-Oct-18 24(35)

Amplitude Shift Keying (ASK)• Switch on-off the carrier:

• Signal representation:

• Is it similar to something?

• Signal spectrum?

( ) ( ) ( )cos 2c c

x t A m t f t= π

( ) 1 or 0m t =


Lecture 9

binary ASK:

general case: a fixed number of levels


Lecture 6 10-Oct-18 25(35)

Amplitude Shift Keying (ASK)

• Signal spectrum (FT):

• Square-wave message:

• Random message PSD:

• How to detect?

( ) ( ) ( )2

c

x m c m c

AS f S f f S f f= − + +

0

20

( ) ( ),

1 1sinc ,

2 2 2 2

m n

n

j n

nb

S f c f nf

n Rc e f

T

+∞

=−∞

π

= δ −

= = =

∑

L.W

. C

ouch

II,

Dig

ita

l an

d A

nalo

g C

om

munic

ation S

yste

ms,

Pre

ntice

Ha

ll, 2

00

1.

Lecture 9


Lecture 6 10-Oct-18 26(35)

Binary Phase Shift Keying

• BPSK signal representation:

where is bipolar message.

• Another form of the BPSK signal ->

( ) ( )cos ( )c c

x t A t m t= ω + ∆ϕ⋅

( ) 1m t = ±

( ) cos cos

sin ( )sin

c c

c c

x t A t

A m t t

= ∆ϕ ω −

− ∆ϕ⋅ ω


Lecture 9


Lecture 6 10-Oct-18 27(35)

Binary Phase Shift Keying

• Digital modulation index:

• Important special case (random message)

• Compare with ASK!

• How to detect?

2

d

∆ϕβ =

π

1d

β =


Lecture 9

PSD


Lecture 6 10-Oct-18 28(35)

Frequency Shift Keying

• Discontinuous FSK: ( )( )

( )1 1

2 2

cos , mark (1)

cos , space (0)

c

c

A tx t

A t

ω + θ=

ω + θ

Not popular (spectral noise + PLL problems)


Lecture 9


Lecture 6 10-Oct-18 29(35)

Frequency Shift Keying

• Continuous FSK:


( ) ( )0

cos

t

c cx t A t m d

= ω + ∆Ω τ τ

∫

Lecture 9


Lecture 6 10-Oct-18 30(35)

Appendix 2:

Autocorrelation and Power/Energy Spectral

Density

This material was covered in ELG3175 and ELG3126. You are

advised to consult your notes and the textbook of that course.

A good textbook to follow is this:

L.W. Couch II, Digital and Analog Communication Systems, 7th

Edition, Prentice Hall, 2007. (other editions are OK as well)


Lecture 6 10-Oct-18 31(35)

Power and Energy

• Power Px

& energy Ex

of signal x(t) are:

• Energy-type signals:

• Power-type signals:

• Signal cannot be both energy & power type!

• Signal energy: if x(t) is voltage or current, Ex

is the energy dissipated in 1 Ohm resistor

• Signal power: if x(t) is voltage or current, Px

is the power dissipated in 1 Ohm resistor.

2( )

xE x t dt

∞

−∞

= ∫21

lim ( )2

T

xT

T

P x t dtT→∞

−

= ∫

xE < ∞

0xP< < ∞

Lecture 3


Lecture 6 10-Oct-18 32(35)

Energy-Type Signals (summary)

• Signal energy in time & frequency domains:

• Energy spectral density (energy per Hz of bandwidth):

• ESD is FT of autocorrelation function:

2 22 1( ) ( ) ( )

2x x x

E x t dt S f df S d

∞ ∞ ∞

−∞ −∞ −∞

= = = ω ω

π∫ ∫ ∫

2( ) ( )

x xE f S f=

( ) ( ) ( ) ( )x x

R x t x t dt E f+∞

∗

−∞

τ = − τ ↔∫ ( )0x xR E=

Lecture 3


Lecture 6 10-Oct-18 33(35)

Power-Type Signals: PSD

• Definition of the power spectral density (PSD) (power per

Hz of bandwidth):

• where is the truncated signal (to [-T/2,T/2]),

• and is its spectrum (FT),

( ) ( )2

( )lim

T

x x xT

S fP f P P f df

T

∞

−∞→∞

= ⇒ = < ∞∫

Lecture 3

( ), / 2 / 2 ( ) ( )

0, otherwiseT

x t T t Ttx t x t

T

− ≤ ≤ = Π =

( )Tx t

{ }( ) ( )T TS f FT x t=

( )TS f


Lecture 6 10-Oct-18 34(35)

Power-Type Signals

• Time-average autocorrelation function:

• Power of the signal:

• Wiener-Khintchine theorem :

( ) ( ) ( )1

lim2

T

xTT

R x t x t dtT

∗

−→∞

τ = − τ∫

( ) ( ) { }( )x x x x

P P f df P f FT R∞

−∞

= ⇒ = τ∫

( ) ( )21

lim 02

T

x xTT

P x t dt RT −→∞

= =∫

Lecture 3


Lecture 6 10-Oct-18 35(35)

Periodic Signals

• Power of a periodic signal:

• Autocorrelation function:

• Power spectral density (PSD):

221( ) (0)x n x

Tn

P x t dt c RT

∞

=−∞

= = =∑∫

( ) ( ) ( ) 021 jn

x nTn

R x t x t dt c eT

∞

ω τ∗

=−∞

τ = − τ = ∑∫

( ) { }2

( )x x n

n

nP f FT R c f

T

∞

=−∞

= τ = δ −

∑

Lecture 3

( ) 0jn tnn

x t c e

+∞

ω

=−∞

← = ∑

Prove these

properties!

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